On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity

If the constitutive law linking the second Piola-Kirchhoff stress tensor and the right Cauchy-Green strain tensor derives from a potential, then the Cauchy stress tensor and the logarithm of the left Cauchy-Green strain tensor are linked by a related potential. We give a new and concise proof which is based on an explicit formula expressing the derivative of the exponential of a tensor

Variance of the virtual displacement

We show that the affine structure of the 3-dimensional space is deeply enmeshed with the static laws expression. The relevance of tensorial rules for calculus in mechanics is thus enhanced. The virtual work principle is stated precisely but a little twist is given to the usual statements: the tensorial nature of the so-called virtual displacement vector is asserted to be covariant

Conditions de compatibilité en mécanique des solides

Thèse d'EtatThe compatibility conditions, associated with partial differential equation of deformable bodies, are used as a guideline of the present work. The basic idea, first presented by Darboux in the context of the general theory of surfaces, consists in replacing the Christoffel symbols by vectors called the Darboux vectors. These vectors are related to rotations in a similar way as the instantaneous rotation vectors in rigid body dynamics. The compatibility conditions are revisited here in the framework of large strain. Two systems of decoupled partial differential equation allow to obtain the displacement of the deformed body by successive integrations. Our results show the validity of the developed tools. An original investigation of three-dimensional Riemann manifolds, with the same curvature as a sphere, is carried out. The theory of surfaces is also studied by introducing the Darboux vectors. A surface is rebuilt from his two fundamental forms in accordance with the Bonnet theorem. The particular study of a minimal surface leads to an efficient building process from the knowledge of the boundary. A new concept, called sister minimal surface, is introduced and its application is developed in the case of two examples. Finally the equivalence between the cancellation of the Riemann-Christoffel curvature tensor in a shell and the Gauss-Codazzi-Mainardi conditions on its mean surface is established. Further developments of the present work would be concerned with the rigid body, treated as a six-dimensional Riemann manifold.Le fil conducteur est celui des conditions de compatibilité des systèmes aux dérivées partielles de la Mécanique des Solides Déformables. L'idée initiale, présentée dans l'ouvrage de Gaston Darboux sur la théorie générale des surfaces, est reprise. Elle consiste à remplacer les symboles de Christoffel par des vecteurs appelés vecteurs de Darboux. Ces vecteurs sont associés à des rotations de la même manière qu'un vecteur rotation instantanée est mis en évidence lors de l'étude du mouvement d'un solide rigide. Les conditions de compatibilité en grandes déformations sont ainsi revisitées à la Darboux. Deux systèmes aux dérivées partielles découplés permettent d'obtenir le déplacement du milieu déformé en deux intégrations successives. L'étude de la nature tensorielle des objets exhibés montre la validité de nos concepts. Une étude inédite des variétés riemanniennes de dimension 3 de même courbure que la sphère est développée. De même, la théorie des surfaces est revue en introduisant les vecteurs de Darboux. La reconstruction d'une surface connaissant ses deux formes fondamentales est proposée conformément au théorème de Bonnet. L'étude particulière d'une surface minimale conduit à un processus de construction effectif à partir de la connaissance du bord. La notion de surface minimale sœur est dégagée, deux exemples sont présentés. Enfin l'équivalence entre l'annulation du tenseur de courbure de Riemann-Christoffel dans une coque et les conditions de Gauss-Codazzi-Mainardi sur sa surface moyenne est établie. Des perspectives, regardant le solide rigide comme une variété riemannienne de dimension 6, sont évoquées

Construire une coque mince connaissant ses deux formes fondamentales

International audienceAccording to Bonnet's theorem, it is possible to reconstruct the parametric equations of the surface initially characterised by its two fundamental forms. Starting from the right polar decomposition and the induced compatibility equations the previous theorem was reformulated. This choice is not innocent since it shows a decoupling of the compatibility equations and this new methodology introduce expressions depending on the two fundamental forms. The attainment of the surface requires two successive integrations and the integration steps are thus simplified. An illustration of this approach is presented in the case of two compatible spheric fundamental forms.Il est possible de reconstituer les équations paramétrées d'une surface qui initialement était caractérisée par ses deux formes fondamentales. C'est ce qu'affirme le Théorème de Bonnet. Nous allons le prouver en utilisant comme point de départ la décomposition polaire droite et les équations de compatibilités qui en découlent. Ce choix n'est pas anodin car il fait apparaître un découplage des équations de compatibilité et cette nouvelle mise en scène introduit des expressions qui dépendent explicitement des deux formes fondamentales. L'obtention de la surface nécessite deux intégrations successives et les étapes d'intégrations à effectuer en sont simplifiées. Une illustration de cette démarche est présentée sur le cas de deux formes fondamentales sphériques compatible

The 3é hyperelastic model applied to the modeling of 3D impact problems

International audienceLainé et al. [Nonlinear isotropic constitutive laws: choice of the three invariants, convex potentials and constitutive inequalities, Int. J. Eng. Sci. 37 (1999) 1927-1941.] have proposed a new third order hyperelastic model, named here the 3é model. The present paper is devoted to the modeling of finite deformations of hyperelastic bodies described by the 3é model under contact/impact conditions. A total Lagrangian formulation is adopted to describe the geometrically nonlinear behavior. A first order algorithm is applied to integrate the equations of motion. For the finite element implementation, an explicit expression of the tangent operator is derived. Efficiency and accuracy of the resulting method are illustrated on a numerical example

Pontryagin calculus in Riemannian geometry

International audienceIn this contribution, we study systems with a finite number of degrees of freedom as in robotics. A key idea is to consider the mass tensor associated to the kinetic energy as a metric in a Riemannian configuration space. We apply Pontryagin's framework to derive an optimal evolution of the control forces and torques applied to the mechanical system. This equation under covariant form uses explicitly the Riemann curvature tensor. This contribution is dedicated to the memory of Claude Vallée (1945-2014)

Pontryagin calculus in Riemannian geometry

International audienceIn this contribution, we study systems with a finite number of degrees of freedom as in robotics. A key idea is to consider the mass tensor associated to the kinetic energy as a metric in a Riemannian configuration space. We apply Pontryagin's framework to derive an optimal evolution of the control forces and torques applied to the mechanical system. This equation under covariant form uses explicitly the Riemann curvature tensor. This contribution is dedicated to the memory of Claude Vallée (1945-2014)

Protocol for a partially nested randomised controlled trial to evaluate the effectiveness of the scleroderma patient-centered intervention network COVID-19 home-isolation activities together (SPIN-CHAT) program to reduce anxiety among at-risk scleroderma patients

International audienceObjective: Contagious disease outbreaks and related restrictions can lead to negative psychological outcomes, particularly in vulnerable populations at risk due to pre-existing medical conditions. No randomised controlled trials (RCTs) have tested interventions to reduce mental health consequences of contagious disease outbreaks. The primary objective of the Scleroderma Patient-centered Intervention Network COVID-19 Home-isolation Activities Together (SPIN-CHAT) Trial is to evaluate the effect of a videoconference-based program on symptoms of anxiety. Secondary objectives include evaluating effects on symptoms of depression, stress, loneliness, boredom, physical activity, and social interaction.Methods: The SPIN-CHAT Trial is a pragmatic RCT that will be conducted using the SPIN-COVID-19 Cohort, a sub-cohort of the SPIN Cohort. Eligible participants will be SPIN-COVID-19 Cohort participants without a positive COVID-19 test, with at least mild anxiety (PROMIS Anxiety 4a v1.0 T-score >= 55), not working from home, and not receiving current counselling or psychotherapy. We will randomly assign 162 participants to intervention groups of 7 to 10 participants each or waitlist control. We will use a partially nested RCT design to reflect dependence between individuals in training groups but not in the waitlist control. The SPIN-CHAT Program includes activity engagement, education on strategies to support mental health, and mutual participant support. Intervention participants will receive the 4-week (3 sessions per week) SPIN-CHAT Program via video-conference. The primary outcome is PROMIS Anxiety 4a score immediately post-intervention.Ethics and dissemination: The SPIN-CHAT Trial will test whether a brief videoconference-based intervention will improve mental health outcomes among at-risk individuals during contagious disease outbreak